Some Tests of Permutation Symmetry

Abstract

The two-sample sign test is viewed as a test for the permutation symmetry of a bivariate distribution, and extensions to $k$-variate distributions are sought. Friedman's rank test, [1], although originally intended as a substitute for the $F$-test in a two-way classification, is such an extension. Study of the family of two-sample sign tests obtained by comparing the $k$ coordinates pairwise has yielded a statistic with an asymptotic Chi-square distribution from which a further test of symmetry can be constructed. The statistic is based on more degrees of freedom than Friedman's and is sensitive to a greater variety of alternatives. This extension is analogous to that obtained by Terpstra [2] from the Wilcoxon test. In this case, however, the limiting distribution turns out to be non-singular. The argument leading to the test is not restricted to the case of complete symmetry but may be carried through with any specified degree of asymmetry. The coordinates may also be compared $m$ at a time, $2 \leqq m \leqq k$. The argument can be extended and, with a slight modification, includes the derivation of Friedman's test. Thus a hierarchy of tests of permutation symmetry are available: Friedman's test corresponds to the case, $m = 1$; when $m = k$, the corresponding test turns out to be Pearson's Chi-square.